TO THE EDITOR:

We read with great interest the article of Nagao et al. (1). The authors applied 3-dimensional fractal analysis for quantifying cerebral blood flow (CBF) distribution in patients with probable Alzheimer’s disease (AD) and in healthy volunteers. Imaging studies have shown spatial and temporal heterogeneity of brain structure and function whenever studied. A key question of these studies is how to differentiate abnormal alterations from normal perfusion heterogeneity. Postprocessing (such as fractal analysis) of the SPECT images may help with image interpretation when visual and traditional approaches fail.

Mandelbrot (2) introduced the word fractal from the Latin fractus (“to break”) to describe the finer irregular fragments that appear when objects are viewed at higher and higher magnifications. Practically, the object or process is considered fractal if its small-scale form appears similar to its large-scale form (such as vermis of cerebellum, bronchial tree of lungs, and daily heart rate variability). Highly recursive and self-similar structures and processes without a well-defined shape can be characterized by fractal analysis. Fractal analysis of the image property is based on nonlinear equations and can be described by a power-law equation showing how a property L(ε) of the system depends on the scale ε: 1 where A is the scaling constant and α is the exponent (3).

The authors concluded that the fractal dimension correlated well with cognitive impairment, as assessed by neuropsychologic tests (1). This conclusion is true. However, their decision that CBF distribution becomes more heterogeneous when AD progresses is wrong. The overall heterogeneity (or complexity) of CBF distribution lessens when disease progresses from the moderate to the end stages. This is a definitive question of the exponent α in Equation 1.

The power-law scaling describes how the property L(ε) of the system depends on the scale ε at which it is measured (Eq. 1). The fractal dimension D describes how the total number of voxels M(ε) of the brain SPECT data depends on the scale ε (cutoff level), namely: 2 where k is a constant. This cutoff threshold method is useful for determining the fractal dimension. However, to use this approach, we must derive the relationship between the fractal dimension D and the scaling exponent α (3). We equate these 2 powers of the scale ε (Eqs. 1 and 2) and then solve for the fractal dimension D (3). For 3-dimensional surface rendering of SPECT data, the solution of the fractal dimension is (3): 3 The result is that the authors have reported the value of −α instead of D, leading to their conclusion that “… CBF distribution becomes more heterogeneous when AD progresses in the moderate and end stages” (1).

Strictly speaking, the 3-dimensional fractal dimensions (Eq. 3) for patients with clinical dementia rates of 0, 1, 2, and 3 are 2.48, 2.37, 2.23, and 1.57, indicating more homogeneous CBF when disease progresses. The decrease in the fractal dimension is associated with impairment of the patient’s condition, as was previously found in patients with dementia of the frontal lobe type using 2-dimensional fractal analysis of SPECT perfusion data (4).